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Quantum Criticality:

Competing Ground States in Low Dimensions

Subir Sachdev†

Department of Physics, Yale University, P.O. Box 208120,

New Haven, CT 06520-8120, USA

† URL: http://pantheon.yale.edu/˜subir; E-mail: subir.sachdev@yale.edu

Published in Science 288, 475 (2000).

This cond-mat version has updated references and new footnotes [60, 61, 62, 63,

64].

ABSTRACT

Small changes in an external parameter can often lead to dramatic qualitative

changes in the lowest energy quantum mechanical ground state of a correlated

electron system. In anisotropic crystals, such as the high temperature super-

conductors where electron motion occurs primarily on a two-dimensional square

lattice, the quantum critical point between two such lowest energy states has non-

trivial emergent excitations which control the physics over a significant portion

of the phase diagram. Non-zero temperature dynamic properties near quantum

critical points are described using simple theoretical models. Possible quantum

phases and transitions in the two-dimensional electron gas on a square lattice are

discussed.

1

1 Introduction

Quantum mechanics was originally developed by Schrodinger and Heisenberg as a theory of

non-relativistic charged particles interacting via the Coulomb force, and successfully applied

to a simple two-particle system like the hydrogen atom. However, among its most important

applications has been the description of ∼ 1023 particles found in macroscopic matter. The

earliest example of this was the Sommerfeld-Bloch theory of electronic motion in metals, and

its refined formulation in Landau’s Fermi liquid theory [1]. Although solving Schrodinger’s

wave equation for 1023 interacting electrons appears an impossibly daunting task, Landau

outlined a powerful strategy, involving the concept of “quasiparticles”, which allowed an

essentially exact description of the low temperature (T ) properties of metals. Extensions of

Landau’s approach have successfully described many other phases of matter: the superfluid

phases of 4He and 3He, the superconductivity in metals which is described by the Bardeen-

Cooper-Schrieffer theory, and the quantum Hall liquid state of electrons in two dimensions

in a strong magnetic field. However, in the last decade, attention has been lavished on

new transition metal compounds for which no successful quasiparticle-like theory has yet

emerged for much of the accessible temperature range. The most important among these

compounds are ceramics, like YBa2Cu3O7, in which the electronic motion occurs primarily

in two-dimensional CuO2 layers, and which display “high temperature” superconductivity.

In this article we shall describe a new approach to the collective dynamical properties of

electrons which turns out to be especially useful in two dimensions: the approach focuses

on the notion of competing ground states and its implications for the dynamics of excited

states at non-zero temperatures. Before describing this further, let us review the essence of

Landau’s strategy. His starting point is the proper identification of the quantum ‘coherence’

or order in the ground state of the system. In the theory of metals, the order is that implied

by the distribution in the occupation number of plane wave states of electrons—the plane

waves with small wavevectors are fully occupied, but there is an abrupt decrease in the

average occupation number above a certain “Fermi wavevector”; in the superfluid state of

4He, the order in the ground state is the presence of the Bose-Einstein condensate—the

2

macroscopic occupation of 4He atoms in the ground state. Landau then proceeds to describe

the low energy excited states, and hence the finite temperature properties, by identifying

elementary excitations which perturb the order of the ground state in a fundamental way.

These excitations can be thought of as new, emergent particles (or ‘quasiparticles’) which

transport spin, charge, momentum, and energy, and whose mutual collisions are described

by a Boltzmann-like transport equation. In metals, the quasiparticles are electrons and holes

in the vicinity of the Fermi wavevector, while in 4He they are phonon and roton excitations.

The systems we shall consider in this article are delicately poised between two or more

distinct states with very different quantum ordering properties and low-lying excitations.

The energies of the states are quite close to each other, and only at very low temperatures is

a particular one picked as the ground state—at these temperatures Landau’s quasiparticle

approach can apply. However, for somewhat different parameters, it is possible that a dif-

ferent state will be picked as ground state, and again Landau’s quasiparticle approach will

apply at very low temperatures: a crucial point is that the nature and physical properties

of these quasiparticles will, in general, be very different from the previous ones. At slightly

higher temperatures it is impossible to ignore the competition between the different states

and their respective quasiparticles: the simple quasiparticle picture breaks down, and very

complex behavior can result which is not characteristic of any one of the possible ground

states.

We describe this intricate temperature dependence by the following strategy. Imagine

following the true ground state of the system as a function of some parameter in the Hamil-

tonian, g. It should be possible to find a critical value g = gc such that the ground state

undergoes a quantum phase transition [2, 3] from one possible state for g < gc to another,

with distinct quantum order, for g > gc. We first develop a theory for the ground state

for the quantum critical point precisely at g = gc. In general, this is a difficult task, but

for “second order” quantum transitions, the critical point has special symmetry properties

which often allows significant progress—we will see examples of this below. Empowered with

this knowledge of the physics at intermediate coupling, we move away from the critical point

3

and map out the physics for nonzero |g − gc| and temperature. It should be emphasized

that it is often the case that the point g = gc is in a regime that cannot be experimentally

accessed; however, this does not rule out application of our strategy—it is still useful to

describe the physics at the inaccessible point g = gc, and then use it as a point of departure

to develop a systematic and controlled theory for an accessible value of g.

Our discussion has so far been rather abstract; we will now spell out concrete details by

considering a number of examples of increasing complexity and discussing their relationship

to experimental observations.

2 Ising chain in a transverse field

This is the simplest theoretical model of a quantum phase transition and many key concepts

emerge from its study [3]. It is described by the Hamiltonian (J > 0, g > 0)

HI = −J∑j

(gσxj + σzj σ

zj+1

). (1)

Here σx,zj are Pauli matrices which measure the x, z components of the electron spin on a

magnetic ion in an insulator. The ions reside on the sites j of a one-dimensional chain. Each

site has two possible states | ↑〉j and | ↓〉j, which are eigenstates of σzj with eigenvalues +1

and −1, and thus identify the electron spin on site j as ‘up’ or ‘down’. The two terms in

HI represent different physical effects–the second term prefers that the spins on neighboring

ions are parallel to each other, while the first allows quantum tunneling between the | ↑〉jand | ↓〉j states with amplitude proportional to g.

For g � 1 and for g � 1, the ground states of HI are simple, and the quasiparticle picture

does describe the low T dynamics [4]. For g � 1 we can neglect the quantum tunneling

and the ground state either has all spins up or all spins down. The order in this state is

evident–all the spins are parallel to each other. The quasiparticles are domain walls which

perturb this order—a quasiparticle state, |Qj〉, between sites j and j + 1 has the following

wavefunction: all spins at and to the left (right) of site j (j + 1) are up (down) (see Fig 1).

4

For g = 0 every such spin configuration is an energy eigenstate and therefore stationary;

for small but finite g the domain walls become mobile (and acquire zero point motion). A

theory for the quantum kinetics of these particles, describing their collisions, lifetime, and

the relaxation of the magnetic order, can be developed following Landau’s general strategy.

In the opposite limit, g � 1, we see from Eq 1 that the ground state can be built out of

eigenstates of σxi with eigenvalue +1: these are

| →〉j =1√2

(| ↑〉j + | ↓〉j) , (2)

or a ‘right’ pointing spin, which quantum mechanically is just a linear superposition of up and

down spins. The ground state has all spins pointing to the right, and it is evident that such

a state is very different from the g = 0 ground state, as the two states form distinct quantum

superpositions of the available states in the Hilbert space. The distinction extends also to the

excited states: we define a left pointing spin by the analog of (2), | ←〉j = (| ↑〉j− | ↓〉j)/√

2,

and the quasiparticle states, |Qj〉, now represent a single ‘left’ pointing spin at site j in a

background of ‘right’ spins, rather than a domain wall (see Fig 1). For g =∞ these states are

stationary, but for g < ∞ the quasiparticles develop dynamics; a theory for this dynamics

can again be formulated in the spirit of Landau, and this describes relaxation phenomena at

low T .

We now allow competition between the distinct orders at small and large g by considering

values of g of order unity. Consider first T = 0. It is known that there is a quantum phase

transition between these states at g = gc = 1, i.e. the ground state qualitatively similar to

the g = 0 ground state for all g < 1, while a state like the g =∞ ground state is favored for

g > 1. The ground state precisely at g = gc is very special: it cannot be characterized by

any such simple cartoon pictures. Its fundamental property is one of scale invariance, as is

apparent from the ground state correlation function [5]

〈σzj σzk〉 ∼1

|j − k|1/4 for large |j − k| : (3)

this power-law decay has the property that the functional form of the correlation is only

modified by an overall prefactor if we stretch the length scale (i.e. perform a scale transfor-

5

mation) at which we are observing the spins. In other words, it is not possible by to tell by

an examination of the ground state wavefunction how far apart any pair of well-separated

spins are. At T > 0, a new time scale does appear, namely h/kBT , and a fundamental

property of the quantum critical point of HI is that this time scale (involving nothing but

the temperature and fundamental constants of nature) universally determines the relaxation

rate for spin fluctuations. This is made more precise by examining the zero-momentum

dynamic response function

χ(ω) =i

h

∑k

∫ ∞0

dt⟨[σzj (t), σ

zk(0)]

⟩eiωt, (4)

where σzj (t) is an operator at time t in the Heisenberg picture, and [, ] represents a quantum

commutator. The arguments above and simple dimensional considerations following from

Eq 3 imply that for low temperatures χ obeys

χ(ω) ∼ T−7/4ΦI(hω/kBT ), (5)

with ΦI a universal response function; if, for example, we added a small second neighbor

coupling to HI , the critical coupling gc would shift slightly but Φ would remain exactly the

same. The exact result for ΦI is known, and it is an excellent approximation to just replace

its inverse by a low frequency expansion [3] ΦI(hω/kBT ) = A(1− iω/ΓR + . . .)−1; here A is

a dimensionless prefactor, and we have the important result that ΓR = (2 tan(π/16))kBT/h.

This is the response of an overdamped oscillator with a relaxation rate determined only

by temperature itself [6]. Although a quasiparticle description of this response function is

strictly not possible, we can visualize the dynamics in terms of a dense gas of the |Qj〉particles scattering off each other at a rate of order kBT/h; however, a picture in terms of

the ‘dual’ |Qj〉 particles would also be valid. It is quite remarkable that the strength of the

underlying exchange interaction between the spins does not appear in these fundamental

dynamic scales.

We can use these above results to sketch a crossover phase diagram in the g, T plane.

This is shown in Fig 1. Note that “quantum criticality”, characterized by responses like Eq 5

holds over a range of values of g at non-zero temperature [7].

6

It can be shown that the physics of the quantum Ising model in spatial dimension d = 2

is very similar–quantum criticality is again characterized by Eq 5 (but the exponent 7/4 is

replaced by a different universal numerical value). Similar behavior applies also to quantum

transitions in d = 3 systems with quenched disorder [8]. However, for the analog of HI in

d = 3 (and for all d > 3), the physics of the quantum phase transition is very different [3]—

the kinetic theory of the analog of the |Qj〉 quasiparticles applies even at the critical point,

and their scattering cross-section depends on the magnitude of the microscopic interactions.

Quantum transitions in this class have been studied elsewhere, and have important physical

applications [9, 10, 11]: our discussion of quantum criticality will not apply to them.

3 Coupled ladder antiferromagnet

We turn to a model in d = 2 which is indirectly related to microscopic models of the high tem-

perature superconductors. We consider the antiferromagnet described by the Hamiltonian

[12] (J > 0, 0 < g ≤ 1)

HL = J∑i,j∈A

Si · Sj + gJ∑i,j∈B

Si · Sj (6)

where Si are spin-1/2 operators on the sites of the coupled-ladder lattice shown in Fig 2,

with the A links forming “two-leg ladders” while the B links couple the ladders. There is

a quantum phase transition in HL at a critical value g = gc ≈ 0.3 which is similar in many

respects to that in HI .

We begin by describing the well-ordered ground states on either side of gc and their

respective, low T quasiparticle theories.

For g close to unity we have the magnetically ordered “Neel” state in Fig 2A. This is

analogous to the ground state of HI for small g, with the difference that the mean moment

on the sites has a staggered sublattice arrangement. There is also an important difference in

the structure of the excitations because HL has the symmetry of arbitrary rotations in spin

space, in contrast to the discrete spin inversion symmetry of HI . Consequently the low-lying

quasiparticle excitations are spin waves corresponding to a slow precession in the orientation

7

of the staggered magnetic order. The precession can be either clockwise or anti-clockwise,

and so there is two-fold degeneracy to each spin-wave mode. Because of infrared singular

scattering of thermally excited spin waves in d = 2, the theory of spin-wave dynamics has

some subtleties [7, 13]; nevertheless, the results remain within the spirit of the quasiparticle

picture.

For small g, the ground state is a quantum paramagnet, and a caricature is sketched in

Fig 2B. The average moment on each site has been completely quenched by the formation

of singlet bonds between neighboring spins. This state is similar in many respects to the

large g ground state for HI . It requires a finite energy, ∆, to create quasiparticle excitations

by locally disrupting the singlet order (the analog of flipping a spin for HI): the singlet

bond between a pair of spins can be replaced by a triplet of total spin S = 1 states, and

the motion of this broken bond corresponds to a three-fold degenerate quasiparticle state

(to be contrasted with the two-fold degenerate spin wave above). A conventional quantum

Boltzmann equation can be used to describe the low temperature dynamics of these triplet

quasiparticles [14, 3].

We sketch the crossover phase diagram in the g, T plane in Fig 3 [7] following Fig 1.

For g ≤ gc, quantum criticality appears for ∆ � kBT � J . Here, dynamic spin response

functions have a structure very similar to that described near Eq 5: the relaxation rate

ΓR continues to be proportional to kBT/h, but now only approximate results for the pro-

portionality constant are available [15, 3]. If we describe the dynamics in the basis of the

triplet quasiparticles, then these results imply that the scattering cross-section is universally

determined by the energy kBT alone [14]. As we lower kBT across ∆ (for g < gc), this

scattering cross-section evolves as a function of the dimensionless ratio ∆/kBT alone, and

for very low T is determined by ∆ alone. One remarkable consequence of this universal

cross-section is that transport coefficients, like the spin conductance σs (which determines

the spin current produced by the gradient in an applied magnetic field), are determined by

fundamental constants of nature [16] and the ratio ∆/kBT :

σs =(gµB)2

hΦσ

(∆

kBT

). (7)

8

Here g is the gyromagnetic ratio of the ions carrying the spin, µB is the Bohr magneton,

and Φσ is a universal function with no arbitrariness in either its overall scale or in that of

its argument. Note that, well into the quantum critical region, σs is proportional to the

universal number Φσ(0), and so is determined by constants of nature alone.

Although it is certainly not appropriate to take HL as a literal model for the high tem-

perature superconductors, it is notable [17] that many measurements of spin fluctuations in

the last decade display crossovers which are very similar to those found in the vicinity of

the quantum critical point in Fig 3. We take this as evidence that the high temperature

superconductors are near a quantum critical point whose spin sector has universal properties

closely related to that of HL [18]: a specific microscopic calculation, involving competition

between the states to be discussed in Section 4, which realizes such a scenario was presented

in [19]. The evidence has appeared in the following experiments: (i) the dynamic spin

structure factor measured in neutron scattering experiments [20] at moderate temperature

obeys scaling forms similar to Eq 5; (ii) as we discuss in Fig 4, crossovers in the nuclear spin

relaxation rate [21, 24, 25] as a function of carrier density and temperature match very well

with the spin dynamics of the different regimes in the g, T plane in Fig 3; (iii) low tempera-

ture neutron scattering measurements [26] at higher carrier density show a resolution-limited

peak above a finite energy gap—this is a signal of the long-lived triplet quasiparticles, like

those found at low T for g < gc in HL; such a peak was argued early on [15] to be a generic

property of the vicinity of a quantum critical point, like that in HL, proposed for the high

temperature superconductors [18]. A further test of quantum criticality in the spin fluctu-

ations could be provided by measurements of the spin conductance and comparison with

Eq 7, but such experiments have not been feasible so far.

4 Electronic ground states in two dimensions

We have so far discussed simple models of quantum phase transitions whose physics is now

well understood. Here we turn to more realistic models of the high temperature supercon-

9

ductors. As we mentioned in Section 1, electronic motion in these materials occurs primarily

in two-dimensional CuO2 layers. The Cu ions are located on the vertices of a square lattice,

and it is widely believed that only the dynamics on a single 3d Cu orbital is relevant, with

the occupation numbers of the other orbitals being inert. So we are led to consider a simple

tight-binding model of electrons with a single orbital on every site of a square lattice, along

with Coulomb interactions between the electrons. If the electron density is precisely unity

per site, then the ground state is known to be an insulator with Neel order (this corresponds

to the state in Fig 2A at g = 1) for the range of parameters found in the stoichiometric com-

pound La2CuO4. It is possible to vary the electron density in the square lattice by doping

such a compound to La2−xSrxCuO4, and then x measures the density of holes relative to the

insulating state with one electron per site. High temperature superconductivity is found for

x greater than about 0.05.

Much theoretical work in the last decade has addressed the physics of this square lattice

model for small x. We will discuss various proposals for ground states, with an emphasis on

finding sharp distinctions between them i.e. distinguishing states which cannot be smoothly

connected by the variation of a parameter in the Hamiltonian, and which must be separated

by a quantum phase transition. Often, the theoretical debate has been about different

approximation schemes to computing properties of states which are ultimately equivalent;

we avoid such issues here—indeed, we advocate that a sound approach is to use a theory for

quantum critical points, separating distinct ground states, to develop a controlled expansion

at intermediate coupling.

A minimal approach to identifying possible ground states is to assume that they are fully

characterized by broken symmetries of the underlying Hamiltonian i.e. a simple electron

mean-field theory of the broken symmetry properly identifies the elementary excitations

(however, as discussed above, this does not rule out highly non-trivial quantum critical

points whose excitations control the physics over a wide region of the phase diagram). The

symmetries which leave the Hamiltonian invariant (and so may be broken by the ground

state) are time-reversal, the group of spin rotations, the space group of the square lattice,

10

and the electromagnetic gauge symmetry related to charge conservation. Even in this limited

framework, the possibilities are remarkably rich, and it is entirely possible that they will

provide an explanation for all the experiments. More exotic ground states have also been

proposed, and we will note them briefly in a subsection below.

One important state has already made an appearance in our discussion in Section 3, and

is known to be the ground state x = 0: the Neel state sketched in Fig 2A. It is apparent by

a glance at the staggered arrangement of spins in Fig 2A that we can view this state as a

density wave of spin polarization at the wavevector K = (π/a, π/a), where a is the square

lattice spacing. For small x 6= 0, spin density waves with a period incommensurate with the

underlying lattice have been observed [27]: these states have a mean spin polarization at a

wavevector K which varies continuously away from (π/a, π/a).

The other ground state of central importance is, of course, the superconducting state.

This is formed by Bose condensation of electrons in Cooper pairs, which leads to the breaking

of the elctromagnetic gauge symmetry. It is known that the pair wavefunction has the sym-

metry of the dx2−y2 orbital in the relative co-ordinate of the two electrons. Recently, interest

has focused on the question of whether the pair wavefunction is on the verge of acquiring

an additional imaginary component with dxy (or possibly s) symmetry: such an instability

would also break time-reversal symmetry [28, 29, 30, 31]. It has been argued [31] that the

quantum phase transition between two such superconductors could very naturally explain

the quantum criticality, similar to the scaling form (5), observed in recent photoemission

experiments [32].

A state which makes a frequent appearance in theoretical studies is one with “Peierls”

order [60, 61]. In models with half-integral spin per unit cell, such order was argued [33] to

be a generic property of any state reached by a continuous quantum transition with restores

the broken spin rotation symmetry of a Neel state. The Peierls order is associated with

broken translational symmetries, and examples are shown in Fig 5: in these states all sites of

the square lattice are equivalent, but links connecting nearest neighbor sites spontaneously

can acquire distinct values for their charge and energy densities (and therefore, for the mean

11

value of the exchange coupling 〈Si ·Sj〉). We also considered a quantum transition restoring

spin rotation symmetry in Section 3, and mentioned its relevance to the NMR measurements

in Fig 4; however, the issue of spontaneous Peierls ordering did not arise there because the

links were already explicitly inequivalent in the Hamiltonian HL in Eq 6. It is believed

[33, 15, 34] that the universal spin fluctuation properties in the vicinity of the quantum

critical point of Section 3 apply also to cases where spontaneous Peierls order appears in the

paramagnetic state. Evidence for the spontaneous Peierls ordering in Fig 5A has emerged

in numerical studies [35, 36] of square lattice models at x = 0 and with first and second

neighbor hopping of electrons, and also in nearest neighbor hopping models for x > 0 [37].

A competitor state to Peierls order for the quantum paramagnet is the “orbital antiferro-

magnet” [38, 39, 40]: this state breaks time-reversal and translational symmetries, but spin

rotation symmetry and the combination of time-reversal and translation by an odd num-

ber of lattice spacings remains unbroken. There is a spontaneous flow of electrical currents

around each plaquette of the square lattice, with clockwise and anti-clockwise flows alter-

nating in a checkerboard pattern–see Fig 6. Ivanov et al. [41] have proposed that a closely

related state (in their formulation, there are strong fluctuations of the orbital currents, but

no true long-range order) is responsible for the “pseudo-gap” phenomenology of the high

temperature superconductors—the pseudo-gap is the partial quenching of low energy spin

and fermionic excitations at temperatures above the superconducting critical temperature.

For our final conventional state, we consider a charge density wave. Much recent ex-

perimental work has centered around the discovery of charge ordering in certain high tem-

perature superconductors and related materials [42]. An especially stable state, observed

for x ≈ 1/8, has a charge density wave at wavevector K = (π/(2a), 0); depending upon its

phase, the charge density wave can be either site-centered or bond-centered, as shown in

Fig 7. Current experiments do not distinguish between these two possibilities. Site-centered

ordering was considered in some early theoretical work [43, 44, 45], although with a very

different charge distribution than is now observed. Bond-centered ordering has also been

considered [46, 47, 19, 60], and has some attractive features: it enhances singlet-bond for-

12

mation between spins, optimizing the energy gained through quantum fluctuations in an

antiferromagnet, and so is preferred by the same effects that led to the Peierls ordering in

Fig 5. Also, bond-centering is naturally compatible with the observed coexistence of charge-

ordering and superconductivity at lower temperatures [48], while site-centering is expected to

lead to insulating behavior. Let us also mention that superposition of charge density waves

with different non-collinear K can lead to an insulating Wigner crystal state; this could be a

Wigner crystal of holes, or with an even number of particles per unit cell, a Wigner crystal

of Cooper pairs [46].

Clearly, a fascinating variety of phase diagrams and quantum phase transitions are possi-

ble among the states we have discussed above. In principle, many of the order parameters can

co-exist with each other, and this adds to the zoo of possibilities. Future experiments with

increased sensitivity should make it possible to more clearly detect more of these orderings,

and thus select between various scenarios.

Spin-charge separation

Finally, we discuss more exotic possibilities of states which cannot be completely charac-

terized by ordering discussed above. Of particular interest has been the early proposal of

Anderson and others [49, 50] that there could be an insulating state with spin-charge separa-

tion i.e. the electron falls apart (‘fractionalizes’) into separate deconfined excitations [51, 52]

which carry its spin and charge. A fundamental property of these deconfined phases is that

superconducting states in their vicinity allow low energy vortex excitations with quantized

magnetic flux equal to hc/e [53, 54, 55, 62]: the elementary flux quantum is always hc/2e,

but it can be argued quite generally that core energy of a hc/e vortex is lower than twice that

of a hc/2e vortex. The kinetic energy of the superflow well away from the vortex core always

prefers the smaller flux hc/2e, and so requiring global stability for hc/e vortices becomes

a delicate question of balancing core and superflow contributions [53, 62]. Nevertheless, it

is possible in principle that a magnetic flux decoration, flux noise, tunneling, or other ex-

periment could observe metastable or stable hc/e vortices or vortex-anti-vortex pairs: this

13

would be a “smoking-gun” signal for deconfinement [63].

Another important class of experiments [56] measure the response to non-magnetic im-

purities like Zn or Li, and these could also provide clear-cut answers on the nature of the

order parameter and on issues of confinement. These impurities substitute on the Cu site,

and so are directly within the plane of the two-dimensional electron gas. Such deformations

are very effective in disrupting the quantum coherence of the ground state, and so serve

as effective probes of its structure. For example [57, 58], replacing only 0.5% of Cu by Zn

dramatically broadens the peak in the neutron scattering cross-section [26], arising from the

triplet quasiparticles discussed in Section 3. Further interesting developments are sure to

follow [64].

5 Conclusion

Correlated electron systems in two dimensions are in a privileged position. Those in three

dimensions either form good Fermi liquids or ordered states with order parameters like those

in Section 4: in the latter case, quantum fluctuations of the order parameter are weak and do

not lead any unusual non-quasiparticle behavior even at zero temperature phase transitions

[11]. In contrast, in one dimension, quantum fluctuations of the order parameters are so

strong that they usually preclude the emergence of long-range order, and so quantum phase

transitions are harder to find. It is in two dimensions that there is a delicate balance

between order and fluctuation, and a host of interesting quantum critical points, with non-

trivial universal properties, can appear between different competing orders. This article

has considered some simple examples of the dynamical properties of systems near such a

point. The phases on either side of the critical point are usually amenable to a quasi-particle

description at low enough temperatures. However, a key point is that the quasiparticle states

are very different for the two phases: so at slightly higher temperatures when both phases

can be thermally excited, neither quasiparticle description is appropriate. Instead special

scale-invariance properties of the critical point have to be used to develop a new framework

14

for finite temperature dynamics.

The availability of a large number of two-dimensional correlated electron systems (in-

cluding the high temperature superconductors), along with the highly non-trivial theoretical

framework necessary to describe them, makes this one of the most exciting research areas

in condensed matter physics. As we have already noted, the increased sensitivity of future

experiments, including neutron scattering, tunneling , magnetic resonance, photoemission

and optics, along with better sample preparation techniques, will surely uncover much new

physics. Many interesting theoretical questions, on the classification of ground states and

quantum critical points, and on the description of dynamical crossovers in their vicinity, re-

main open. The interplay between theory and experiment promises to be mutually beneficial,

in the best traditions of physics research.

15

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18

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[59] I thank Matthew Fisher, Chetan Nayak, T. Senthil and Matthias Vojta for useful dis-

cussions. This research was supported by US NSF Grant No DMR 96–23181.

[60] The spin-Peierls state of Fig 5A is also a bond-centered charge-density wave with

wavevector K = (π/a, 0); the charge density on all sites is equivalent in this state, but the

charge density on the bonds is not.

[61] A state with spin-Peierls (or bond-centered “stripe”) order co-existing d-wave-like su-

perconductivity, was predicted early on in S. Sachdev and N. Read, Int. J. Mod. Phys. B 5,

219 (1991) (this paper is available online at http://onsager.physics.yale.edu/p34.pdf; see

especially Fig 5); fundamental reasons [33] for the existence of the spin-Peierls order in un-

doped and doped antiferromagnets were also given. Evidence for precisely such co-existing

orders is given by McQueeney et al. Phys. Rev. Lett. 82, 628 (1999) in La1.85Sr0.15CuO4;

the symmetries of Fig. 5 of McQueeney et al. and Fig. 5A here are identical. I thank S.

Kivelson for drawing my attention to this paper.

[62] The results of [53] (and related work) have been reviewed in S. Sachdev,

Int. J. Mod. Phys. B 6, 509 (1992). An online version of this paper is at

http://onsager.physics.yale.edu/c9.pdf. We add some remarks here related to recent de-

velopments: (i) With the modern consensus on the meaning of spin-charge separation

(S. A. Kivelson and D. S. Rokhsar, Phys. Rev. B 41, 11693 (1990); [54]), spin and charge

are separated in a BCS superconductor. So the title of the paper should more properly read

“Stable hc/e vortices in a superconductor near a normal state with spin-charge separation”.

19

This change in terminology does not modify the discussion of the physical properties of the

various phases. (ii) The mean-field theory presented in the paper ignores the compactness

of the U(1) gauge field ~a, and the related effects of instanton fluctuations. Such effects will

not modify the structure of the phases, but will make the total ~a-flux in vortices uncertain

modulo 2π ([51]; S. Sachdev and M. Vojta, cond-mat/9910231). From the first equation

in (17) in the paper, this implies that only the value n∆(mod 2) is physically significant.

[63] Also notable is the proposal of Senthil and Fisher, cond-mat/0006481,

cond-mat/0008082. In the formulation of [62], the “vison” excitation of Senthil and Fisher

is the vortex in Normal State II with with n∆(mod 2) = 1; the b boson is not condensed

in this phase, and so there is no significance to the value of nb. In the superconductor,

finiteness of the free energy F is imposed by (17,18) in [62], and this requires that the vison

must trap a flux of an odd multiple of hc/2e. The free energy F of [62] displays the same

“flux regeneration effect” in a cylindrical geometry under the protocol described by Senthil

and Fisher, and represents a simple way of understanding their effect. Indeed, this effect

requires the enhanced stability of hc/e vortices discussed in [53, 62], for otherwise hc/2e

vortices generated by thermal or quantum fluctuations as one moves into the normal state

will annihilate the flux in the center of the cylinder. So a direct search for hc/e vortices,

proposed in [53] and reviewed in [62], represents an alternative experimental route. Some

related comments have been made recently by P. A. Lee and X. G. Wen, cond-mat/0008419

and M. Franz and Z. Tesanovic, cond-mat/0002137 cond-mat/0010436.

[64] For an updated review of experiments and theories of non-magnetic impurities in the

high temperature superconductors, see S. Sachdev and M. Vojta, cond-mat/0009202.

20

Figures

g

T

gc0

Domain-wallquasiparticles

Flipped-spinquasiparticles

Quantumcritical

Figure 1: Phase diagram of HI . The quantum phase transition is at g = gc, T = 0, and

the dashed red line indicates a crossover. Quasiparticle dynamics applies in the blue shaded

regions: for g < gc the quasiparticle states are like the |Qj〉 states, while for g > gc they are

like the very different |Qj〉 states. The quantum critical dynamics in the pink shaded region

is characterized by Eq 5.

21

(B)

(A)

= -( )/ 2(C)

Figure 2: The coupled ladder antiferromagnet–the spin-1/2 degrees of freedom, Si, reside on

the blue circles. The A links are the full red lines and have exchange J , while the B links

are dashed lines and have exchange gJ . The Neel ground state for g > gc appears in (A).

The paramagnetic ground state for g < gc is schematically indicated in (B). The ellipses in

(B) represents a singlet valence bond, (| ↑↓〉 − | ↓↑〉)/√

2 (shown in (C)), between the spins

on the sites.

22

g

T

gc

0

Tripletquasiparticles

Spin waves

Quantumcritical

Figure 3: Crossover phase diagram [7] for HL with the same conventions as Fig 1. The ground

state is a paramagnet (Fig. 2B) for g < gc and the energy cost to create a spin excitation,

∆ is finite for g < gc and vanishes as ∆ ∼ (gc − g)zν where zν is a critical exponent. There

is magnetic Neel order at T = 0 for g > gc (Fig. 2A) and the time-averaged moment on any

site, ~N0, vanishes as g approaches gc from above. Quasiparticle-like dynamics applies in the

blue shaded regions. For g < gc, in the cartoon picture of the ground state in Fig 2B, the

triplet quasiparticle corresponds to the motion of broken singlet bond in which Fig fig2C is

replaced by one of | ↑↑〉, | ↓, ↓〉, or (| ↑↓〉 + | ↓↑〉)/√

2. For g > gc, the quasiparticles are

spin-waves representing slow, long-wavelength deformations of the ordered state in Fig 2A.

23

0

2000

4000

6000

8000

10000

0 200 400 600 800 1000

La2-x

SrxCuO

4

1/ T

1 (se

c -1 )

T (Kelvin)

x = 00.020.040.0750.1150.15

Figure 4: Measurements [21] of the longitudinal nuclear spin relaxation (1/T1) of 63Cu

nuclei in the high temperature superconductor La2−xSrxCuO4 as a function of x and T . This

quantity is a measure of the spectral density of electron spin fluctuations at very low energies.

At small x, 1/T1 increases rapidly as T is lowered (see red circles): this is also the behavior in

the spin-wave regime of Fig 3 (g > gc)—the energy of the dominant thermally excited spin-

wave decreases rapidly as T decreases, and so the spin spectral density rises [22]. In contrast,

at large x, 1/T1 decreases as T is lowered (see blue squares): this corresponds with the triplet

quasiparticle regime of Fig 3 (g < gc)—the low energy spectral density is proportionally to

the density of thermally excited quasiparticles, and this becomes exponentially small as T

is lowered. Finally, at intermediate T , 1/T1 is roughly temperature independent for a wide

range of T (see orange triangles) and this is the predicted behavior [18, 23] in the quantum

critical regime of Fig 3.

24

(A)

(B)

Figure 5: Two examples of square lattice ground state with Peierls order. All sites are

equivalent, and distinct values of the energy and charge densities on the links are represented

by distinct colors. These distinctions represent a spontaneous breaking of the symmetry of

the square lattice space group. The spontaneous ordering appears because it optimizes

the energy gained by resonance between different singlet bond pairings of near neighbor

spins. This figure should be contrasted with Fig 2 where there is no spontaneous breaking

of translational symmetry, and the distinction between the links is already present in the

Hamiltonian Eq 6. 25

Figure 6: The orbital antiferromagent. The gradient in the red shading represents the

direction of spontaneous current flow on the links, which breaks time-reversal symmetry.

26

(A)

(B)

Figure 7: Charge density waves with (A) site- and (B) bond-centering; both states have a

4 × 1 unit cell. The state in (A) is symmetric about reflections in a vertical axis running

through the red or green sites, while that in (B) requires a vertical axis centered on the bond

between two red sites or two green sites. The colors of the sites represent different charge

densities. Spin ordering can also be present for an appropriate K but is not shown. Note that

the bond-centered ordering naturally suggests an effective model for the spin fluctuations

much like the ladder model of Section 3: the spins of Fig 2 reside on the red sites of (B)

(with no spins on the green sites) and the weaker gJ exchange interactions (represented by

the dashed lines Fig 2) extend across the green sites.

27